'\ci\-'if^-iy'^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1314 ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE By K. Wieghardt and W. TiUmaim Translation of ZWB Untersuchungen und Mitteilungen Nr. 6617, November 20, 1944 NACA Washington October 1951 ..NTSDEPAR-mEMT ;,ARSTON SCIENCE UBRARY RO. BOX 117011 QiiuS>. GAINESVILLE. FL 3^t>i ^VS iC i^^HO ^iiir'ir NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 131^ ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE* By K. Wieghardt and W. TilLmann Abstract: As a supplement to the UM report 6603, measurements in tur- bulent friction layers along a flat plate with 'rising pres- sure are further evaluated. The investigation was performed on behalf of the Aerodynamischen Versuchsanstalt Gottingen. Outline: 1. SYMBOLS 2. INTRODUCTION 3. TEST SETUP k. TEST RESULTS 5. ON THE GRUSCHWITZ CALCULATION METHOD 6. ON AN ENERGY THEOREM FOR FRICTION LAYERS 7. SUMMARY 8. REFERENCES 1. SYMBOLS X position rearward from leading edge of the plate y distance from wall u velocity component in x-direction V velocity component in y-direction U^V velocity components outside of the friction layer p density \i viscosity \, of the air- V kinematic viscosity p static pressure "Zur tvirbulenten Reibungsschicht bei Druckanstieg." Zentrale fiir wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluft- zeugmeisters (ZWB), Berlin-Adlershof, Untersuchungen und Mitteilungen Nr. 6617, Kaiser Wilhelm-Institut fiir Stromungsforschung, Gottingen, November 20, 19iiU. NACA TM ISl^^ P 2 q = 2U = p + q dynamic pressure total pressure D ? Q = ^ U dynamic pressure outside of the friction layer Ux/v U62/V & Reynolds number Reynolds number of the friction layer friction layer thickness 6-, = / (1 - 77)^ displacement thickness ^0 ^6 62 = ^3 = u momentum loss thickness dy energy loss thickness %2 - ^l/S^ Hop — Oo/Oq ^ form parameters of the velocity profile T) = 1 - "u(S2)" U ^•f' = T turbulent shearing stress wall shearing stress /p p j/p U"^ local friction drag coefficient mixing length 2. INTRODUCTION For the calculation of laminar boundary layers in two-dimensional incompressible flow, numerous calculation methods have been developed NACA TM 131^ on the basis of Prandtl's boundary -layer equation; in contrast, only the semiempirical method of E. Gruschwitz (reference l) and its improvements by A. Kehl (reference 2) and A. Walz (references 3^ ^, and 5) are avail- able for turbulent friction layers. The reason, as is well known, lies in the lack of a mathematical law for the apparent shearing stress t which originates by the turbulent mixing of momentum. Prandtl's expres- sion for the mixing length so far has led to success only in cases where a sufficient niJinber of correct (and also sufficiently simple) data on the variation of the mixing length can be obtained directly from the geometry of the flow as for instance in the case of the free jet. 'As a basis for the Gruschwitz method there serve, therefore, besides the momentum equa- tion for friction layers, three statements which are partly empirical and lack a theoretical basis. They are as follows: A. The velocity profiles in the turbulent friction layer for vari- able outside pressure form, after having been made adequately dimension- less, a single-parameter family of ciirves, if one disregards the laminar sublayer; thus, every profile may be characterized by a single quantity (t^) . B. A differential equation, likewise derived solely from experi- ments, concerning the variation of this parameter in flow direction as a function of the pressure variation and of the momentum thickness. C. An assumption concerning the wall shearing stress. Gruschwitz inserts a constant as first approximation. A. Kehl (reference 2) then improved the Gruschwitz method. According to his measurements which extended over a larger Re-number range than those of Gruschwitz, it is necessary to insert in statement B, for a higher Re-number of the friction layer, a function of the Re-number U52/V instead of a certain constant b. Furthermore, Kehl obtained better agreement between the calculation and his test results by sub- stituting in scatement C for the wall shearing stress the value which results for the respective Re(52) number at a flat plate with constant outside pressure. Finally, A. Walz (references 3^ ^) and 5) greatly simplified the integration method mathematically so that for prescribed variation of the velocity outside of the friction layer U(x) the mom- entum thickness, the form parameter t\, and hence the point of separ- ation can be calculated very quickly. No statement is obtained regarding the wall shearing stress, since it had, on the contrary, been necessary to make the assumption C concerning Tq in order to set up the cal- culation method at all. Thus it seemed desirable to investigate the friction drag of a smooth plate for variable outside pressure. On one hand, direct interest in this exists in view of the wing drag; on the other hand, one could NACA TM 131^ expect an Improvement in the above calculation method if an accurate statement regarding Tq could be substituted for the assumption C. In order to arrive at the simplest possible laws, friction layers along a flat smooth plate were investigated where a systematically increasing pressure was produced by an opposing plate. By analogy with the behavior of laminar boundary layers the wall shearing stress was expected to decrease in the flow direction up until separation, more strongly than In case of constant outside pressure. Instead, Tq increased, after a certain starting distance, more or less suddenly to a multiple of the initial value. A brief report on this striking behavior of the friction drag has already been published (reference 6) . In the present report, these tests are further evaluated and compared with those of Gruschwitz and Kehl. Considerable deviations result in places; however, it was not possible to develop a better calculation method with this new test mate- rial either. Since the application of an energy theorem had proved expedient for the calc\ilation of laminar boundary layers (reference 'j) , a theoretical attempt in this respect was made for turbulent friction layers, too; however, it did not meet with the same success. Merely an interpretation for the statement B can be obtained in this manner, which is. however, not cogent. 3. TEST SETUP The test setup and program have been described in the preliminary report. A new measuring method was developed where with the aid of a pressure rake and of a multiple manometer (reference 8) turbulent friction layers co\ild be measured quickly and accurately and the computational evaluation greatly simplified. Friction layers were measured at p = for two different veloc- ities U = const., four cases, I to IV, with rising, and one, V, with diminishing pressure (figs. 1 to 6) . In cases I and II, p increases almost in the entire measuring range linearly with rearward position with respect to the leading edge of the plate x, whereas in cases III and rv the outer velocity U (U ~ x"^) decreases with a power of x (figs. 1 to 6) . The velocities were about 20 to 60 meters per second; the test section length was 5 meters so that Re -numbers up to 107 were attained. The Re-number of the friction layer (formed with the momentum thickness) increased to from 2 to 7 X lo5. MCA TM IBlij- h. TEST RESTTLTS Mainly the variation of the wall shearing stress had heen described in the brief preliminary report (reference 6); later the measurements were evaluated more thoroughly. First, we plotted figures 1 to 6 for the different pressure variations: the outer velocity U, the local friction drag coefficient c^', the displacemenx thickness 6^, the momentum thickness 82^ and the energy loss thickness 5^ .treated in section 6; in analogy to the momentum loss thickness 82^ 80 is a measure of how much kinetic energy of the flow is lost mechanically due to the friction layer, that is, is converted to heat by the effect of the friction forces. In case of constant outside pressure Cf' depends, for a smooth plate, only on the Re-number. The test points from figure 1 lie between the formulas for cf' according to L. Prandtl (reference 9)^ F. Schultz- Grunow (reference 10 ), and J. Nikuradse (reference 11) which in the test range Ux/v = 3 X lO-' to 10' differ only slightly. For rising pressure, a slight decrease of c^' with rearward position results at first, according to figures 2 to 5j however, after a certain distance c^ ' increases more or less suddenly to a multiple of its original amount and decreases again only at the end of the test section where for test tech- nical reasons the pressure increase could no longer be maintained. W. Mangier (reference 12) found the same unexpected behavior in further evaluating the measurements of A. Kehl (reference 2). In contrast, c^' varies only slightly in case of decreasing pressure. Thus, on one hand, -T*- > must be responsible for the strong increase in wall shearing stress; furthermore, the Re-number of the friction layer is of importance since first a certain starting distance is required. Therefore, we plotted in figure 7 c^' against the dimensionless -r- — ^. Our measure- ■'■ y dx ments resulted in a comparatively narrow bundle of curves; however, the results according to Kehl -Mangier, drawn in in dashed lines, cannot be brought under a common denominator in this manner. The test arrangement of Kehl was more general insofar as he had at first a piece of laminar boundary layer and, moreover, in some cases first a pressure drop, and then an adjoining pressure rise; in our tests, in contrast, the friction layer, starting from the leading edge of the plate, had been made tur- bulent by a trip wire and the pressure increased monotonically. NACA TM 13114- In want of a better criterion, we can see from figure 7 that the strong increase in friction drag is not to be expected as long as ^ ^ < 2 X 10-3 Q dx The displacement and momentum thickness as well as the wall shearing stress represent only a summary of the development of a friction layer. Therefore, we shall consider below the velocity profiles and shearing stress profiles. The variation of — against the rearward position from the leading edge for constant wall distance is particularly illus- trative (figs. 8 and 9)- In case II (fig. 8), — suddenly drops steeply in the layers near the wall whereas it decreases continuously in case IV. Accordingly, the characteristic lengths S]_, 62, and 60 increase at these points more strongly than before. Since the drag coefficient Cf' depends essentially on the variation of ^— , one recognizes at once in ox case II the point of maximum wall shearing stress at x w 3'3 meters. The shearing stresses are obtained from Prandtl's boundary- layer equation which may be transformed with the aid of the continuity equation and of the Bernoulli equation = -UU„ valid outside of the friction p dx ^ layer. With U^ = -p , % = v^, Uy = ^, and Q = ^ U , one obtains S_i_^,li^x_!^ r^^ay (1) Sy 2Q U U U U Jq U The integration which is still to be performed yields additionally a control value for the wall shearing stress — = = - / dy (2) 2Q 2 Jq ^y 2Q However, the value for Tq obtained in this manner is not as reliable as the one calculated from the momentum theorem because here graphical differentiation is applied more often. According to a suggestion by MCA TM 131^ Professor Betz, the momentum theorem may be transformed for the calcvil- atlon of Tq in the following manner (compare reference 6): 2Q y2+Hi2 dx\ -.,) . (h,, - H,,)i 3^ a, (3) where a suitable mean value of E-^Q ^ ^l/^2 ^^ substituted for H2_2- Then the second term is small compared to the first and plays only the role of a correction term so that essentially only one graphical differ- entiation has to be performed. First, the shearing stress profiles are plotted for constant outside pressure in figure 10. The shearing stress t with the appertaining wall shearing stress t^ and the wall distance y with the momentum thickness S2 are made diraensionless. These dimensionless profiles are almost completely identical although a systematic variation with the Re-number is recognizable. The profiles t against y for the two cases II and IV with press\ire rise follow in figures 11 and 13; figures 12 and 1^4- show the corresponding dimensionless shearing stress profiles t/tq against y/52- In the case II where the pressure p increases approximately linearly, the t -profiles differ considerably for various rearward positions, especially the wall distance at which T reaches its maximum value is subject to great changes. In contrast, -0 2T the dimensionless profiles in case IV where U <*< 53(x/l°') ' meters per second is valid are very similar. Only the one profile at X = 1.99 meters stands out sharply without any perceptible reason. According to D. R. Hartree (reference 13) ^ similar velocity and hence also shearing stress profiles result for laminar boundary layer in the case U ~ x~ since for laminar flow t is simply t = U ^. In the turbulent friction layer, the velocity and hence also the shearing stress profiles thus vary in this case; however, the latter at least can be described - although somewhat forcibly - by a single -parameter family of curves, according to figure ik. A. Buri (reference 1^) attempted to interpret the shearing stress profiles in general as single -parameter family of curves. Bvurl selected for the parameter Tg^ the computed wall tangent of the dimensionless S2/St' shearing stress profile ?„ = — \^—] because for this quantity a ^o\ay/y^o relation may be read off immediately from the boiindary-layer equation. 8 NACA TM 131^ For y=0, u=v=0 and therewith r„ = ^ which yields ^ To dx -^ -— I = -^. The tangent to the shearing stress profile thus calcu- ^y/y=o ^"^ lated is known in general to fit very hadly the shearing stress variation in the proximity of the wall determined from measurements. In the tur- bulent friction layer, the velocity u decreases only in immediate wall proximity - in the so-called lajninar sublayer - to zero so that the tan- gent direction deviates from the above calculated value even for very small wall distances. This is shown also in figure 15, which represents the shearing stress profiles in wall proximity and the pertaining wall tangents found by calculation; for constant outside pressure, too, the curves against y apparently begin to drop from the point y = out- ward with a definite angle rather than continue with a horizontal tangent (fig. 10). Nevertheless, Fg^ could be used at first as a com- puted quantity for the characterization of a definite shearing stress profile. However, in case II, for instance, the same shearing stress profile would have to be present for x = 3-19 meters and for X = 3-79 meters according to figure 15, which is obviously not true according to figure 12. In general, the shearing stress profiles could not be characterized by one other parameter alone either. At least two quantities would be required for this, for instance, the magnitude and the dimensionless wall distance of the maximum ^jj^^^/tq. Finally, we calculated from the shearing stress the mixing lengths 5u according to Pranatl's expression t = p2 -— oy and plotted them in Sy figures I6 to 19 against the wall distance y or in dimensionless form Z/62 against y/52. Like the thickness of the friction layer, 2 increases more and more with further rearward movement; in case IV, 2 even increases on approaching the end of the test section to 30 milli- meters. The diminishing of the mixing length for large wall distances cannot be specified, as is well known, since 2 there is computed as the quotient of two small quantities, namely, t and r— • Here again I increases linearly in wall proximity. In the tube, the result had been 2 = O.^J-y and at the plate for constant pressure, according to Schult^ -Grunow (reference 10) 2 = 0.i4-3y (dashed in figs. I7 and 19)- Here, at rising pressure, I increases at first again linearly but far more rapidly. In case II, 2 attains the maximum value 2 = l.ly and in case IV even 2 = 2.0y. Although no fixed relation exists between 2/y in wall proximity and the wall shearing stress, it is striking in figures I7 and 19 that 2/y is largest just for those rearward positions where the wall shearing stress Tq also (compare figs. 3 and 5) attains its maximum value. NACA TM 131^ "i. ON THE GRUSCHWITZ CALCULATION METHOD Our test material enables us to examine the basis of the Gruschwitz method enumerated under section 1. The assumption A according to which all turbulent velocity profiles concerned form a single -parameter family of curves is again confirmed in figure 20. Here a few velocity profiles of the different cases I to V are plotted for which the evaluation accidentally had resulted in exactly the same ratio H2^2 ~ ^1/^2^' ^^^ profiles u/U against y/Sg with equal H12 ^^^ hence equal t\ (compare fig. 27) are in agreement even for different velocities^ rear- ward positions^ and pressure variations, thus also after different "previous history." This single -parameter quality covers, however, only the "visible" turbulent part of the profile; the velocities in the laminar sublayer may have a different distribution even for equal t]; otherwise, a unique connection would necessarily exist between the wall tangent Fg^ and i^ or Hj_2 which is certainly not the case according to what was said above . The change of the velocity profiles characterized by the param- eters r| or H2^2 with change in the rearward position is represented in figures 21 and 22 for all measuring series. The weak but throughout systematic dependence of the profile on the Re -number for constant out- side pressure is remarkable; Niku_adse (reference 11), on the other hand, obtained here always the same profile with t] = 0.515 and H;L2 = 1-302. In order to apply the momentum theorem for the Gruschwitz method, the relation between H]_2 and t), necessarily unique for a single- parameter profile class, must be known. The results for all profiles of our measuring series are plotted in figure 23- All of them lie below the curve indicated by Gruschwitz but the majority of Kehl's points also lie below the original Gruschwitz curve. Since, however, all the results do not greatly deviate from one another, the assumption A may be regarded as correct in good approximation. The long dashed curve was calculated by Pretsch (reference I5) for power profiles; the power pertaining to a certain H-L2 1^ given in the figure at the right. Concerning the short dashed curve, compare section 6. Masters are different for assumption B. Gruschwitz had obtained from his test evaluation &P dg(6p) o -, — — —^-^ == ari - b, with a = 8.9^+ x lO'-^ and b = h.6l x 10"^ Q dx 10 NACA TM I31U 8(^2) ^^ "^^^ total pressure at the wall distance 52: 2 6(52) = P + |[^(52)] Because of tj = 1 - u(52) /U and the Bernoulli equation outside of the friction layer p + Q = constant, we may also write dg(62) dp -, d(Qii) = — P + Q(l - il) = dx dx"- -■ dx Kehl, who investigated a larger Re-number range, found b to be addi- tionally dependent on U52/v; therefore, he plots 52 dg(S2) 52 d(QTi) b = ari ^^ = a.T\ + {h) ' Q dx ' Q dx against U52/V • This has Deen done for our measurements in figure 2h. In this diagram, Gruschwitz obtained a horizontal straight lihe _-> b = constant = ii.6l X 10 and Kehl the plotted curve which, starting o from U52/V = 2 X 10 slowly drops. These two lines have been drawn solidly as far as measurements existed. Our measurements show that at least for the cases I and II - that is, for linear steep pressure rise - b can be assumed neither as constajit nor as a unique function of U62/V since we obtain in this diagram two essentially deviating cuj.ves. The same result is obtained also in th^ cases III and IV for higher Re-numbers. Only for U62/V < 10 , the deviations of the measuring points may be interpreted as scatter. It is not the Kehl relation but the simpler Gruschwitz relation which is confirmed here. For U52/V < 2 X 10^ Kehl took the drawn variation of b in order to obtain significant calculation results directly behind the transition point from laminar to turbulent flow. The two points with U62/V < 10^ m the case V are not an argu- ment against this b-variation for the reason that the friction layer had been made artifically t\irbulent to start with by means of the trip wire. Below, we shall attempt a theoretical interpretation of the relation B which so far has been set up and investigated in a purely empirical manner . MCA TM 131^ 11 Since the variation of the wall shearing stress proved to be very complicated, according to figiires 2 to 6, we cannot improve upon assxjmp- tion C which refers to the wall shearing stress. In order to estimate the effect of this assumption on the calculation, the cases II and IV were calculated with the aid of Walz ' s simplified integration method under different assumptions regarding c^' . Figure 25 shows the result for the momentum thickness 52- In the momentum theorem, we may put ^12 ~ constant since It appears only in one term 2 + H]_2 so that even great variations of H^^g ^^^ °^ relatively small importance. In case II, one obtains better agreement (at least up to x = 3^) between calculation and tests by using the expression for c^ ' obtained as a function of U52/V for the plate without pressure rise than by putting c^' = constant. Conversely, the agreement for case IV is better with the assumption c^' = constant. In view of the actual variation of Cf ' which varies from case to case, no generally valid rule can there- fore be set up for it, even before the region of the steep rise of Cf ' . The results of the r\ calculation are represented in figure 26. The three different calculation methods worked out by Walz are based on the following assumptions: Gruschwitz-Walz b = constant = k.6l X 10"3 and c„' = constant = k x 10"3 Gruschwitz-Walz b = constant = ij-.6l X 10"^ and Cf.' = 0.025l(U52/v)"^A Gruschwitz-Kehl-Walz b = 0.01614- 0.85 log (U&2/V) U52/V - 300 and = 0.0251(U52/V)"^/^ 12 NACA TM I31U As was to be expected according to figure 2U, the last, most complicated method is precisely the one that shows the greatest deviations compared to the experiment. In our cases II and IV, the first and simplest cal- culation method is the best. The agreement between that calculation and the test is still good even in the region of steep rise Cf ' . From there on, however, large differences result. 6. ON AN ENERGY THEOREM FOR FRICTION LAYERS Like the approximation methods for laminar boundary layers, the Gruschwitz method is based on Karman's momentum equation which is obtained by integration of Prandtl's boundary-layer equation with respect to the wall distance y. One obtains thereby a statement on the total momentum loss of the friction layer. In analogy, a statement on xhe total energy loss in the friction layer caused by the friction can be obtained if the boundary -layer equation pUUjj + OVUy = -Pj^ + Ty = PUU^ + Ty (5) pu. (after addition of -p-(ux + Vy) = because of continuity) is first multiplied by u and then integrated with respect to y i. J%„(| „. . 1 ,2) ,, -.8 pu^v + P ^ / ^x dy = UTy dy and because of 2 pu V I U2v = -H U2 rS I "Y' "- '' J ^v ^y = / ^"^ ^^ = kf p^(l "^ - 1 ^^) ^y (6) This equation signifies that the loss in kinetic energy per unit length transverse to the flow direction in the friction layer equals the rate of doing work of the turbulent shearing stresses. NACA TM 131^ 13 For the laminar boundary layer t = [lUy is valid so that / "^Uy dy = i-t / Uy2 dy here signifies the dissipation, that is, the energy converted to heat per unit time. In turbulent friction layers, in contrast, no simple relation between the shearing stress and the velocity profile exists; if it did, the essen- tially single -parameter family of tiirbulent velocity profiles woiild have to include also a single -parameter profile class of the shearing stresses which is not the case according to the test evaluation. If we define, corresponding to the momentum loss thickness, an energy loss thickness (7) ^0 and the following dimensionless e = work of the shearing stresses _ j _r_ S_ u , o\ work against the wall shearing stress Jq t^ Sy U we can write equation (6) also as e = 1 c^'u3 I^K) (9) Since we have thus obtained one new equation with two new unknowns, this energy theorem at first does not help us any further either. How- ever, we can calculate the energy loss thickness 5o from the velocity distributions and obtain, due to zhe single -parameter quality of the velocity profiles, a fixed relation between the quantities &]_, 62, and 6^. Plotting thus the ratio H02 = B3/S2 against H-]_2 = ^±/^2 ^°''^ ^-^^ profiles measured, we obtain figure 27. All points come to lie, with only very small deviations, on one "street." For power profiles ^/U = (y/S)"^ the long dashed curve AHng Ik NACA TM 1314 vould result with A = U/B and B = l/3- A good fairing curve for the actual profiles at pressure rise and di op results if we put A = I.269 and B = 0.379' On the basis of this empirical relation, 63 may now be calcxilated from 5]_ and 62 with sufficient accuracy. If one more relation could be found for the ratio e as well, it would be possible to calculate from the new equation (9) for instance cj.' . The calculation of e from the measiiring results is made difficult by the fact that it results as the quotient of two quajitities which can both be fo\ind only by graphic differentiation. This explains the great variation of the e-values plotted for the different cases in figure 28. On the whole, e varies only comparatively little; even when c^.' for instajice in case IV increases from 3 "to 16 X 10"^, e remaint between 0.8 and 1.2. At first glance, e seems to have, according to the defining equation (8), the significance of an efficiency which could not exceed 1. Actually, however, e may assume any arbitrary value, according to the velocity and shearing stress profile concerned; in the immediate proximity of tiirb^olent separation, above all, where Tq vanishes, e may assume arbitrary magnitude. Because of the inaccuracy of the calculation from the test data, it was not possible to determine for e a relation to the other boundary- layer quantities. However, one can set up an interesting analogy between the energy equation and Gruschwitz ' s assumption B. If one substitutes the above relation for H02 which was found experimentally in the energy d52 equation and eliminates with the aid of the momentum theorem, one dx obtains U„ -R dH-|p Htp(A - 2e) + 2eB — + ^ = J^ c^' (11) U Hi2(Hi2 - B)(Hi2 " 1) dx 2AHi2(Hi2 " 1) ^ On the other hand, one may differentiate out the Gruschwitz -Kehl relation (assumption B), equation (k) , and v—ite ^ _!_ J^ di] dHi2 ^ b - aT] (12) U 2ti dH]_2 dx 2ii regarding t) as a function of H-]^2- ^ = ^(^12)- We now assume that the relation B follows from the energy theorem. If this is the case. MCA TM I31U 15 the left sides of equations (ll) and (12) must, first of all, be iden- tical. We therefore equate, as an experiment, the left sides of the equations and obtain a differential equation for t\(^E±2) > "the solution of which reads P (%2 - 1)^/^^ ' ^^ "^^^"^"12 2/(1 - B) ^'3^ (H12 - B) From the H32 - H3_2 relation, ve had found for B: B = O.379. If, furthermore, we put, for adaptation to the test results, the number equal to 0.986, the function H-]_2(il) is, accordirig to figure 23, quite well satisfied by equation (l3)- Thus, the left sides of the energy equation (ll) and of the rela- tion B (equation (12)) seem to be identical. Then the right sides also must be equal which - solved with respect to b - results in Hip(A - 2e) + 2eB b = at) + -^ Cj'tI with A = I.269 AHi2(Hio - 1) and B = 0.379 (l^) Therewith, a relation between qaantltiee of the velocity and of the shearing stress profile, based on the energy theorem, has been found for the assumption of the Gruechwitz method. In practice, of course, this equation is of no help, either, since no data concerning the newly intro- duced quantity e are available although one deals here, according to the defining equation (8), with a comparatively illustrative quantity. Thus one may conclude from equation (l4) only that it is improbable that invariably b = constant (as Gruschwitz assumes) or that b depends solely on U52/v (as Kehl presupposes), for b appears here as a func- tion of the respective velocity profile [r\ and Hx2) s-s well as of e and Cf' which likewise is not determined merely by U52/^. However, theoretically this conclusion is not cogent, either. It is, in itself, conceivable that the course of the t'orbulence mechanism is such that, in spite of different previous history, the complicated relations between T^, c^', and e have precisely the properties which cause b to be, for every rearward position, for instance a function of U62/V only and the test evaluation according to figure 2k actually shows that b, at least within a certain region, hardly varies. l6 NACA TM 131^ 7. SUMMARY The report deals with measixrements in the turbxilent friction layer along a flat p]ate where the static pressiire from the leading edge of the plate onward systematically rises or decreases. In case of pressure rise, there results after a certain starting distance, a large increase in wall shearing stress to a m\iltiple of the initial value; this has already been briefly commented on in the report UM Nr. 6603. The further evaluation of the test material (calculations of the shearing stresses and mixing lengths) also gave qualitative information on this problem. As rule of thumb, it 'can merely be said that this strong increase in friction drag does not occur as -rr- -^^ < 2 X 10^ is valid. ^ Q dx Furthermore, the empirical relations on which the Gruschwitz method is based are checked with the aid of the measurements. It is again confirmed that the tiirbulent velocity profiles form with sufficient accuracy a single -parameter family. Gruschwitz obtains from an empirical relation a differential equation for the variation of that parameter in flow direction; Kehl Improved that equation by not fixing a quantity b contained in it as constant, like Gruschwitz, but by considering it as a function of the Re-number of the friction layer. The present measure- ments confirm at first the simple relation b = constant. However, for higher Re-numbers of the friction layer - in the region of strongly increasing wall shearing stresses - different values for b resiilt, according to the previous history of the friction layer; here a relation of the form b = b(U62/v) is no longer sufficient, either. It is shown that this Gruschwitz -Kehl relation can be interpreted as statement of the energy theorem applied to friction layers. However, this energy theorem which simply signifies that the work of the t\irbvilent shearing stresses equals the loss of kinetic energy in the friction layer does not provide any practical help (for instance for setting up a calculation method) . As long as a sufficient statement for calculation of the shearing stresses themselves is lacking, a link between the newly intro- duced total work of the shearing stresses and the known friction layer quantities, -r^ displacement or momentum thickness, etc., also is lacking. For the same reason, it is not even possible to calculate the wall shearing stressj an approximation value for it must be inserted in the Gruschwitz method. Thus the result of the present investigation is. NACA TM 131^ 17 on the whole, negative with respect to the problem of calciilating in advance turbulent friction layers; however, the test material represented in the figures might prove useful for f\irther theoretical considerations. Translated by Mary L. Mahler National Advisory Committee for Aeronautics 18 NACA TM 131^ 8. REFERENCES 1. Gruschwitz, E.: Die tiu-bulente Reibungsschicht in ebener Stromung bei Druckabfall und Lruckanstieg. Ing. -Archiv, Bd. II, Heft 3, September 1931, PP- 321-3^6. 2. Kehl, A. : Untersuchiingen iiber konvergente und divergente, turbulente Reibungsschichten. Ing.-Archiv, Bd. XIII, Heft 5, 19^3, pp. 293-329. (Available as R.T.P. Translation No. 2035, British Ministry of Air- craft Production. ) 3. Walz, A. : Graphische Hilfsmittel zur Berechnung der laminaren und turbulenten Reibungsschicht. Lilientlial-Gesellschaft fur Luftfahrtforsch\ing Bericht S 10, 19^0, pp. h-^-lh. h. Walz, A.: Zur theoretischen Berechnung dea Hochstauftriebsbeivertes von Tragfluge±profilen ohne und mit Auftriebsklappen. ZWB For- scb.ungsbericht Nr. I769, March 19^3- 5. Walz, A.: Naherungsverfahren z\ir Bei echr-'-^ng dei laminaren und turbulenten Reibungsschicht. Untersuchungen \ind Mitteilungen Nr. 3060. 6. Wieghardt, K.: Ueber die Wandschubspannung in turbulenten Reibungs- schichten bei veranderlichem Aussendruck. Untersuchungen \ind Mitteilungen Nr. 6603. 7. Wieghardt, K.: Ueber einen Energiesatz zur Berechnung laminarer Grenzschichten. (Available from CADO, Wright -Patterson Air Force Base, as ATI 33090.) 8. Wieghardt, K.: Staurechen und Vielfachmanometer fur Messungen in Reibungsschichten. Technische Bericht, Bd. 11, No. 7, 19^4, p. 207. 9- Prandtl, L.: Zur turbulenten Stromung in Rohren und langs Flatten. AVA-Ereebn. IV. Lief., 1932, p. I8. 10. Scnultz-Grunow, F.: Neues Reibungswiderstandsgesetz fur glatte Flatten. Luftfahrtforschung, Vol. 17, No. 8, Aug. 20, 19^+0, p. 239- (Available as NACA TM 986.) 11. Nikuradse, J.: Tvurbulente Reibiingsschichten. Publication of the ZWB, 19^2. 12. Mangier, W.: Das Verhalten der Wandschubspannung in turbulenten ReibuTigsschichten mit Druckajistieg. Untersuchungen und Mitteilungen Nr. 3052, 19^3- NACA TM 131^^- 19 13. Hartree, D. R. : On an Equation Occurring in FaUoier and Skan's Approximate Treatment of the Equations of the Boundary Layer. Proc. Cambridge Phil. Soc. , vol. 33, pt. 2, April 1937, PP- 223-239. ik. Buri, A.: Eine Berechnungsgrundlage fur die turhulente Grenzschicht bei beschleunigter und verzogerter Grundstromung. Eidgenossische Technische Hochschule, Zurich, 1931. (Available as R.T.P, Translation No. 2073, British Ministry of Aircraft Production.) 15. Pretsch, J. : Zur theoretischen Berechnung des Profilviderstemdes. Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Avail- able as NACA TM 1009 •) 20 NACA TM I31I+ 0.005 0.004 0.003 0.002 0.00 f 0.006 0.004 0.003 0.002 0.001 X U = const. = 17.8mA 16 2 3 U = const. = 33.0m/^ 4 X [m] 5 16 12 4 X [m] 5 Figure 1,- Constant outer velocity, displacement, momentum, and energy thickness; local drag coefficient. NACA TM I31J+ 21 Figure 2.- Pressure rise: Case I. Displacement, momentum, and energy thickness; local drag coefficient. 22 NACA TM I31U Figure 3.- Pressure rise: Case n. Displacement, momentum, and energy thickness; local drag coefficient. NACA TM 1314 23 Figure 4,- Pressiire rise: Case HI. Displacement, momentum, and energy thickness; local drag coefficient. 2k NACA ™ I31U Figure 5.- Pressure rise; Case IV. Displacement, momentum, and energy thickness; local drag coefficient. NACA OM 131^ 25 30 \ 20 / .+■ U y" ^ •X. 60 p[kg/m^] -"^ 40 r Figure 6.- Pressure drop: Case V. Displacement, momentum, and energy thickness; local drag coefficient. 26 NACA TM I31I+ 4 5 6 Q dx Figure 7.- Cf' against — -^ for pressure rise and pressure drop; comparison with measurements of Kehl, NACA TM 131^^ 27 w w 0) a, W W •rH cd 2^H ■o m o -i-i ^ -t-> :=) • r-t w •rH -o • rH O O 1 — I 0) > I 00 CD flH ^^D 28 NACA TM 1311+ (U w w u a o a ctJ rt 0) w O ■r< •1-1 M •4-' • rH KJ o r-H > I CTJ P4 NACA TM 131^4- 29 U-- 1 7.6 mis • A = 0.4d7m + 1.067m I.9d7m X 4.067m 1.0 0.8 0.6 0.4 0.2 -^ \x. \ X U -33.0 mis • X - 0.467m + 1.087 m ° 1.967m ^ 4.067m N V \ V, ^^x. -4- •- 2 10 " % '" Figure 10.- Dimensionless shearing stress profiles for constant outer velocity. 30 NACA TM I31I+ 5 S ^ E E E t\ r\ i\ f. t\ rv. ~~: c\i C\j o^i ri "O t~^ NACA TfL 131*+ 31 o o W CD I — 1 o w w 0) t^ W ho s Oj OJ w w m -a o •l-i w Q I CO r-l ft 32 NACA TM I31I+ ^ ^ c\] .--j 05 t-j 'SJ ^ O <1 lO ^ (U w ci 0) '? ^6, fn ^ m m (a u a u M OJ r— < O CU 0) d ■ rH 0) CO I CO u NACA TM 131^^ 33 0) w o w w 0) a w a; o w w ho •i-t 0) Si m w w -a o CO a Q ^ 3^ NACA TM 13li^ Case U Case N Figure 15.- Shearing stress profiles and wall tangents (Buri form parameter). MCA TM 131^^ 35 0) w O 0) 1=1 W W Q o 1 — I ■rH I -— I CD h ^iS. 36 NACA TM I31U 0) u a o CO 0) w w 0) o I .>5 to CO o •rH Q NACA TM'131^ 37 0) o 0) w u u w o to I— ( bjO .3 CO |iH 38 NACA TM I31I1 0) ;:! w w CD m o 01 Ui I a; Q I oi r-l a> u (in NACA TM 131^ 39 w 0) r-H O •s o ;^ a. o o 1—1 CO m 0) ■a o -l-l w I o CSl 0) p^ ko NACA TM 1314 ^ nmj ^ I 2 3 Figure 21.- Variation of il for all measirring series. Figure 22.- Variation of H22 for all measuring series. (Significance of point markings, compare fig. 23.) NACA TM I31U kl !> v?o^^5 T3 a a CD CD CD a o I — I CD CO CD U s, k2 NACA TM I31U -O ■♦-^ «i Ci <o «o r>'- c: 1 V, ei ?> ^ Q- :3 ^ 0^ >^ •-H t=l ^^ ^ -. 0^ irj Uj Q) Q) <l) 11 11 •^ ■o -o ^ ::) d c5 cS C3 X + e 9 ■<o <G Qj ^ ■QO <o '^J -O 00 B w T3 r— ( O o to a! X3 f^ NACA TM 131^ 1*3 'O ^ § >s •^ > ^) 1—1 V V, a Qi H w <D Q ^ •^ CO CO •t-l ^ a 'n -M rei • i-H f-t ol > T3 Q) '^ -«-> rt 1 — 1 7i 1 — 1 <a ^ ■^ 'NJ ai •r) (U fn ::! 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